The idea of transporting measures via couplings has
a long history [1]}, [2]},
and has found applications in fluid dynamics, economics, statistics,
machine learning [3]}, [4]}, [5]}, [6]}, and many other fields.
One of the first filtering algorithms to rely on the explicit construction of (optimal) couplings is
the ensemble transform particle filter [7]}, [8]}, which
uses an
importance sampling
approximation of the filtering distribution—the same given by the
bootstrap particle filter [9]}—to
obtain a consistent particle approximation of Monge's optimal
map [2]} (cf. Section ).
Earlier work on particle flows
investigated the use of transport maps
induced by flows of ODEs [11]}, [12]}.
Related ideas include the feedback particle filter
[13]}
and the “Gibbs flow” [14]}.
In particular, [14]} defines an approximation
of the KR rearrangement between an input and a
target measure via the solution of
an ODE whose drift term depends only on the full conditionals of the
target distribution.
This approximation can be used as a proposal density for sequential Monte Carlo (SMC) methods.
Also, the implicit sampling algorithm of [16]}, [17]}
defines a proposal density using an approximate
transport map, which can be evaluated
by
solving a one-dimensional optimization problem: in this case,
optimization is used to define the action of a single map,
not to search for the best rearrangement within a class of candidate maps
(see [18]} for more details).
Among the flow methods,
the variational mapping particle filter implements a particle
flow given by
a particular form of
functional gradient descent on a reproducing kernel Hilbert space [19]}, [20]}.
We clarify its connection with the deterministic map
filter
in Section .
